An infinitely wide, horizontal metal plate lies above a horizonta...

Question

An infinitely wide, horizontal metal plate lies above a horizontal infinite sheet of charge with surface charge density 800 nC/m². The bottom surface of the plate has surface charge density -100 nC/m² . What is the surface charge density on the top surface of the plate?

Accepted Answer

Hello everyone. Let's take a look at this practice problem. A thin nonconducting rectangular sheet is placed over a metal plate having similar surface dimensions and a top surface charge density of 2. micro columns per square meter. What must be the surface charge density at the bottom surface of the metal plate? If the nonconducting sheet has negative 4.6 micro couls per cube per square meter. We have four multiple choice options. Option A zero option B negative 4. multiplied by 10 to the power of negative six Kums per meter squared. Option C negative 2.3 multiplied by 10 to the power of negative six columns per meter squared or option D 4.6 multiplied by 10 to the power of negative six Kums per meter squared. Now, this problem might be a little confusing. So let's draw a picture to make it more clear what's going on. So we have a negatively charged sheet, little negative sign to represent it's negatively charged and it is sitting on top of a metal plate, a conducting metal surface. And we're told that the top of this metal surface has a positive charge and we're given the surface charge density. And the problem is asking us to find the surface charge density at the bottom of the metal plate. Now, for this problem, there are a few important things to consider. Recall that an important law for conductors is that the charge or electric field within the conductor always has a net value of zero. This is because since conductors can freely move their charge around, if there is any influence at all, that will cause there to be some field, it will automatically cancel itself out. In this case. For example, if there is a positive charge and that positive charge at the surface of the conductor, then because it's in such close proximity with a negatively charged object, there should be a negative charge at the bottom of the metal plate because the negatively charged nonconducting plate is pushing electrons within the metal plate down to the bottom. This means that any electric field set up by the nonconducting plate is within the metal plate canceled out by an opposing electric field produced by this charge difference. So the net electric field in the metal plate is zero as is the case with any conductor at all times. Let's make this even more clear with the diagram. You might expect there to be an upwards electric field within the metal plate due to the fact that there is that there are negative charges on the nonconducting plate. And as is convention, we have electric field lines pointing towards negative charges. So you might expect there to be an upwards electric field. But since there are also the positive charges at the top of the metal plate itself and the negative ones at the bottom, the metal plate is creating an opposing downwards pointing electric field, canceling it out. Let's make this more clear with a force equation or the vector equation. So we know that the net electric field within the conductor is zero and its components are a downwards pointing or negative electric field at the top of the conductor. A negative component at the bottom and a positive upward component at the top where the sheet is. So the electric field from the top is negative, the electric field from the bottom of the metal plate is also negative. The electric field from the sheet is positive, it is upwards. Now, since the problem is asking for the surface charge density at the bottom surface, let's take this electric field equation and rewrite it algebraically to solve for e sub bottom, the electric field at the bottom surface of the metal plate adding E sub bottom to both sides of the equation. We find that the electric field at the bottom is equal to the electric field at the sheet eu sheet minus the electric field at the top of the metal plate eu top. So now we have this equation set up for the electric field at the bottom of the metal plate. But we want to find the surface charge density. Now recall a helpful formula recall, the electric field at a sheet of charge is equal to the surface charge density of that sheet divided by two multiplied by the electric constant epsilon knot. Let's apply this formula to each term in our vector equation. So for eu bottom, that's the surface charge density at the bottom of the metal plate, which is what we want to find divided by two epsilon knot. Listen, and we're dealing with magnitudes here and this is equal to the same thing for the sheet. So Sigma sub sheet divided by two epsilon knot minus sigma sub top divided by two epsilon na. Since the two epsilon knot is in all three terms of the equation, these can just get canceled out. And we're left with a simpler formula saying that Mu sub bottom is equal to mu sub sheet or Sigma subs sheet minus sigma sub top. So now we have this relationship between the surface, the three different surface charge densities and sigma sub bottom is the one we want to find. So let's put in the values that the problem gave us. So for the sheet that is a magnitude of 4. multiplied by 10 to the power of negative six Kums per cube per meter squared minus Sigma sub top which is given the magnitude of which is 2. multiplied by 10 to the power of negative six Kums per meter squared. We put this into a calculator or just do the math, we find a magnitude for the surface charge density at the bottom of the metal plate of about 2.3 multiplied by 10 to the power of a negative six cos per meter squared. And so this is the magnitude of the charge at the bottom of the metal plate, the surface charge density. However, as we discussed earlier, we know that this must have been negative since we already know that the charge at the bottom of the plate must be negative. So really our final answer to the problem is negative 2.3 multiplied by 10 to the power of negative six. And if we look at our multiple choice options, we can see that this agrees with option C. So option C is the correct answer to this problem. And that is all for this problem. I hope this video helped you out if it did. And you're and you'd like to see more. Please consider checking out some of our other tutoring videos which will give you more experience with these types of problems. That's all for now and I hope you all have a lovely day. Bye bye.

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An infinitely wide, horizontal metal plate lies above a horizontal infinite sheet of charge with surface charge density 800 nC/m². The bottom surface of the plate has surface charge density -100 nC/m² . What is the surface charge density on the top surface of the plate?

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